Unconstrained Hilbert-series subsequences for Koszul Gorenstein algebras

Construct, for every sufficiently large socle degree d and every permutation π of {1,…,⌈d/2⌉}, an artinian Gorenstein Koszul algebra with Hilbert series ∑_{i=0}^d h_i t^i satisfying h_{π(1)}<⋯<h_{π(⌈d/2⌉)}, thereby establishing that the first half of the Hilbert-series coefficient sequences of such algebras is unconstrained.

Background

The paper proves a Roller Coaster theorem for artinian Gorenstein algebras, showing that arbitrary orderings can occur among the first half of the Hilbert-series coefficients. Its construction uses Nagata idealization of algebras associated with whiskered graphs and produces Gorenstein algebras that are not necessarily Koszul.

The authors obtain a large family of G-quadratic, hence Koszul, Gorenstein algebras failing the weak Lefschetz property, but they do not establish the full Roller Coaster phenomenon within the class of artinian Gorenstein Koszul algebras. The conjecture asks whether the unconstrained behavior can be realized while retaining Koszulness.

References

Let $d \gg 0$ and $\pi$ be a permutation of the numbers ${1, \dots, \lceil d/2 \rceil}$. Then there exists an artinian Gorenstein Koszul algebra with Hilbert series $\sum_{i = 0}d h_i ti$ such that $$ h_{\pi(1)} < \dots < h_{\pi(\lceil d/2 \rceil)}. $$ In other words, the collection of sequences arising from the first half of the Hilbert series of artinian Gorenstein Koszul algebras of large socle degree is unconstrained.

— Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property  (2502.00155 - Holleben et al., 31 Jan 2025) in Conjecture in Section 1 (Introduction); repeated as a concluding conjecture in Section 5